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Creating and probing electron whispering-gallery modes in graphene

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Science  08 May 2015:
Vol. 348, Issue 6235, pp. 672-675
DOI: 10.1126/science.aaa7469
  • Fig. 1 Confined electronic states in microscopic electron cavities defined by pn junction rings in graphene.

    (A) The rings are induced by the STM tip voltage bias (Vb) and back-gate voltage (Vg), adjusted so as to reverse the carrier polarity beneath the tip relative to the ambient polarity. The pn junctions act as sharp boundaries giving rise to Klein scattering of electronic waves, producing mode confinement via the whispering-gallery mechanism. The cavity radius and the local carrier density are independently tunable by the voltages Vb and Vg. Electron resonances are mapped out by the STM spectroscopy measurements (see Fig. 2). Shown are the STM tip potential U(r) and the quantities discussed in the text: the STM tip radius (R), its distance from graphene (d), and the local (μ0) and ambient (μ) Fermi levels with respect to the Dirac point. n and p label the electron and hole regions. (B) Spatial profile of WGM resonances. Confinement results from interference of the incident and reflected waves at the pn rings (dashed lines). The confinement is stronger for the larger angular momentum m values, corresponding to more oblique wave incidence angles. This is illustrated for m = 5 (weak confinement) and m = 13 (strong confinement). Plotted is the quantity Embedded Image, the real part of the second spinor component in Eq. 1.

  • Fig. 2 Confined electronic states probed by STM measurements.

    (A) Differential tunneling conductance (dI/dVb) for a single-layer graphene device, as a function of sample bias (Vb) and back-gate voltage (Vg). The gate map was obtained after increasing the probe-tip work function by exposure to deuterium to shift the interference fringes vertically downward (fig. S5) (17). The two fans of interference features, marked WGM′ and WGM′′, originate from WGM resonances in the DOS (see text). (B) Interference features in dI/dVb, calculated from the relativistic Dirac model. The features WGM′ and WGM′′ in the (Vg,Vb) map originate, respectively, from the conditions ε = μ0 and ε = μ0 + eVb (see text). The boundaries of the WGM′ (and WGM′′) regions are marked by dashed (and dotted) white lines, respectively. arb. units, arbitrary units. (C) dI/dVb spectra taken along the horizontal line in (A) at Vb = 230 mV. (D) dI/dVb spectra taken along the two vertical lines in the map in (A) at Vg = 16 V (red line) and Vg = –11 V (blue line, scaled ×3 and offset for clarity) (see text for discussion). The four peaks at positive bias at Vg = 16 V are fit to Gaussian functions, with the fits shown in the lower right of the figure. The peaks labeled 1′′,2′′,3′′… correspond to WGM resonances probed at the energy ε = μ0 + eVb, whereas the peaks labeled 1′,2′,3′…, are the same WGM resonances probed at the Fermi level ε = μ0, giving rise to the WGM′′ and WGM′ fringes in the gate maps, respectively. The resonance spacing of order 40 mV translates into a cavity radius of 50 nm, using the relation Embedded Image (see text).

  • Fig. 3 Contributions of the WGM resonances with different m to the DOS for the relativistic Dirac model.

    (A) Colored curves represent partial-m contributions from angular momentum values m = 1,2,3,4,5 (see Eq. 3), evaluated for a confining potential Embedded Image with curvature value Embedded Image. Different curves show the partial DOS contributions defined in Eq. 3, which are offset vertically for clarity. The inset shows the total DOS versus particle energy ε and the curvature κ (see text). The black curve shows the total DOS trace along the white line. (B) The Dirac wavefunction for different WGM states (see Eq. 1). Spatial structure is shown for several resonances in the partial DOS (black dashed circles mark the pn junction rings). The quantity plotted, Embedded Image, is the same as in Fig. 1B. The length scale Embedded Image (the same in all panels) is marked. Note the confinement strength increasing with m.

Supplementary Materials

  • Creating and probing electron whispering-gallery modes in graphene

    Yue Zhao, Jonathan Wyrick, Fabian D. Natterer, Joaquin F. Rodriguez-Nieva, Cyprian Lewandowski, Kenji Watanabe, Takashi Taniguchi, Leonid S. Levitov, Nikolai B. Zhitenev, Joseph A. Stroscio

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